Richards was bornEmilio Radocchia in Hartford, Connecticut. He began playing the xylophone at the age of six, and while in tenth grade, he joined the Hartford Symphony Orchestra and performed with various jazz musicians around New England. After serving in the First Cavalry Army Band as assistant band leader, he graduated from the Julius Hartt School of Music (now part of the University of Hartford) and later from Hillard College.

Throughout the 1950s and 1960s, he played various percussion instruments with jazz ensembles first in New York and then in Los Angeles, where he settled permanently in 1959. He collaborated with such artists as Frank Sinatra, Nelson Riddle, Judy Garland, Sarah Vaughan, and Doris Day. In 1962, in response to a request from President John F. Kennedy, Richards and a small jazz band joined Sinatra on a world tour to benefit needy children. It was during this tour that Richards began a what is currently the world's largest collection of percussion instruments from around the world.

After the tour, Richards continued to work with Sinatra and recorded sessions with such artists as the Beach Boys, Jan and Dean, Bing Crosby, and Nat King Cole. In the meantime, Richards contributed to a steadily increasing number of film scores. One of the many film composers he worked with over the years was the late Jerry Goldsmith. In addition to four Star Trek films, Richards and Goldsmith worked together on such films as Seven Days in May, The Sand Pebbles, Planet of the Apes, Patton, Papillon, Chinatown, Logan's Run, Poltergeist, Twilight Zone: The Movie, Gremlins, Innerspace, The 'burbs, Gremlins 2: The New Batch, The River Wild, Congo, Executive Decision, L.A. Confidential, Air Force One, and Mulan.

The many other film scores on which Richards has worked include The Nutty Professor (1963), Doctor Zhivago (1965), Cool Hand Luke (1967), Butch Cassidy and the Sundance Kid (1969), Shaft (1971), The Exorcist (1973), Soylent Green (1973), Jaws (1975), Taxi Driver (1976), All the President's Men (1976), Close Encounters of the Third Kind (1977), The Deer Hunter (1978), National Lampoon's Animal House (1979), Ghostbusters (1984), Indiana Jones and the Temple of Doom (1984), Cocoon (1985), Ferris Bueller's Day Off (1986), Fatal Attraction (1987), Wall Street (1987), Lethal Weapon (1987), Die Hard (1988), The Hunt for Red October (1990), Beauty and the Beast (1991), Batman Returns (1992), Jurassic Park (1993), Mission: Impossible (1996), Independence Day (1996), Men in Black (1997), X-Men (2000), Spider-Man (2002), Chicago (2002), Pirates of the Caribbean: The Curse of the Black Pearl (2003), and Spider-Man 2, just to name a few. He has also worked on countless television series and mini-series and has been part of the orchestra for several Academy Awards ceremonies.